Problem

Find the exact values of $s$ in the interval $[0,2 \pi)$ that satisfy the condition, $\sin s=-\frac{1}{2}$. \[ s= \] (Use a comma to separate answers as needed. Simplify your answers. Type exact answers, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)

Solution

Step 1 :We are asked to find the exact values of \(s\) in the interval \([0,2 \pi)\) that satisfy the condition, \(\sin s=-\frac{1}{2}\).

Step 2 :The sine function has a value of \(-\frac{1}{2}\) at two points in the interval \([0,2\pi)\), namely at \(\frac{7\pi}{6}\) and \(\frac{11\pi}{6}\).

Step 3 :This is because the sine function is negative in the third and fourth quadrants, and the reference angle corresponding to a sine value of \(\frac{1}{2}\) is \(\frac{\pi}{6}\).

Step 4 :Thus, the exact values of \(s\) in the interval \([0,2 \pi)\) that satisfy the condition, \(\sin s=-\frac{1}{2}\) are \(s=\frac{7\pi}{6}, \frac{11\pi}{6}\).

Step 5 :So, the final answer is \(s=\boxed{\frac{7\pi}{6}, \frac{11\pi}{6}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/19111/

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